But, Public Benefit always = Sum of all Private Benefits. You may be thinking direct benefit vs external benefit, but as everyone utility functions are separate their share of the public benefit is different. I specifically mentioned my personal cost (as a share of the public cost) and my personal (indirect) benefit to help clarify the issue, but I think you missed that point.
Cost and benefit are also independent. It possible for a Polio vaccine to provide 100,000 times the public benefit relative to it's cost (eliminating a disease for all time can easily be worth vaccinating people on the other side of the planet). Research can also easily have that magnitude benefit ratio's.
And (public benefits) / (public costs) is only negative if public benefits or public costs are negative you might be thinking Benefits - Costs but that does not really fit the rest of your equations either. Anyway, the idea that public subsidy's should not exceed the ratio of (individual benefit) / (public benefit) sounds good until the public benefit exceeds the cost at which point there is little for an individual to pay the costs.
PS: There is a reason people get PHD's in economics it's a lot more complex than you might think. Just consider adding different levels of diminishing returns to all of those cost benefit equations and your still barely scratching the surface.
Cost and benefit are also independent. It possible for a Polio vaccine to provide 100,000 times the public benefit relative to it's cost (eliminating a disease for all time can easily be worth vaccinating people on the other side of the planet). Research can also easily have that magnitude benefit ratio's.
And (public benefits) / (public costs) is only negative if public benefits or public costs are negative you might be thinking Benefits - Costs but that does not really fit the rest of your equations either. Anyway, the idea that public subsidy's should not exceed the ratio of (individual benefit) / (public benefit) sounds good until the public benefit exceeds the cost at which point there is little for an individual to pay the costs.
PS: There is a reason people get PHD's in economics it's a lot more complex than you might think. Just consider adding different levels of diminishing returns to all of those cost benefit equations and your still barely scratching the surface.